Suppose a researcher is interested in knowing the statistics skills of first-year graduate students 2016 as compared to the skills for those in 2015. It is known that on a statistics test for the first-year grad students in 2015 the population mean is 75 (on a scale of 0-100) with a population standard deviation of 10. To answer the research question, the researcher draw a random sample of first-year grad students in 2016 and administer a comparable statistics test. The researcher is interested in knowing whether the first-year grad students in 2016 perform differently from those in 2015 on the statistics test, and so the researcher wanted to test the null hypothesis H0 : μ = 75. Assume that the researcher chose α = .05 with a two-tailed test, and so the critical values are -1.96 and +1.96.

(1) Suppose that the researcher collected data from a random sample of 100 grad students and found out that the sample mean is 75.5

(a) Assume that the population standard deviation is the same in 2016 and in 2015, that is, the population standard deviation is σ = 10 in 2016. For a sample with n = 100, what is the standard error of the sample mean? Show the steps of your calculation.

(b) What is the test statistic for this random sample in 2016? From comparing the test statistic to the critical values, what decision and conclusion should be made?

(c) Using the sample mean and its standard error (SE), construct a 95% confidence interval (CI) for the sample mean in the form [x ¯ - 1.96 · SE, x ¯ + 1.96 · SE]. Does the CI contain the value 75?